# On q-de Rham cohomology via $$\Lambda$$ -rings

Mathematische Annalen, Feb 2019

We show that Aomoto’s q-deformation of de Rham cohomology arises as a natural cohomology theory for $$\Lambda$$-rings. Moreover, Scholze’s $$(q-1)$$-adic completion of q-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach p-power roots of q, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00208-019-01806-7.pdf

J. P. Pridham. On q-de Rham cohomology via $$\Lambda$$ -rings, Mathematische Annalen, 2019, 1-28, DOI: 10.1007/s00208-019-01806-7